Robust Precoding With Bayesian Error Modeling for ... - IEEE Xplore
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 9, SEPTEMBER 2010
ACKNOWLEDGMENT The authors would like to acknowledge the activity of the Network of Excellence in Wireless COMmunications NEWCOM++ of the European Commission (Contract 216715) that motivated this work.
[21] F. Tufvesson and T. Maseng, “Pilot assisted channel estimation for OFDM in mobile cellular systems,” in Proc. IEEE Veh. Tech. Conf., Phoenix, AZ, May 1997, pp. 1639–1643. [22] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001.
REFERENCES [1] J. A. C. Bingham, “Multicarrier modulation for data transmission, an idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990. [2] Transmission and Multiplexing (TM); Access Transmission Systems on Metallic Access Cables; Very High Speed Digital Subscriber Line (VDSL); Part 2: Transceiver Specification, ETSI TS 101 270-2, Feb. 2001. [3] Digital Audio Broadcasting (DAB); DAB to Mobile Portable and Fixed Receivers, ETS 300 401, Feb. 1995. [4] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb. 1995. [5] R. van Nee, G. Awater, M. Morikura, H. Takanashi, M. Webster, and K. W. Halford, “New high-rate wireless LAN standards,” IEEE Commun. Mag., vol. 37, no. 12, pp. 82–88, Dec. 1999. [6] B. Muquet, Z. Wang, G. B. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixing or zero padding for wireless multicarrier transmissions,” IEEE Trans. Commun., vol. 50, no. 12, pp. 2136–2148, Dec. 2002. [7] H. Steendam and M. Moeneclaey, “Different guard interval techniques for OFDM: Performance comparison,” in Proc. 6th Int. Workshop Multi-Carrier Spread Spectrum (MC-SS), Herrsching, Germany, May 2007, pp. 11–24. [8] T. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [9] L. Deneire, B. Gyselinckx, and M. Engels, “Training sequence versus cyclic prefix—A new look on single carrier communication,” IEEE Commun. Lett., vol. 5, no. 7, pp. 292–294, Jul. 2001. [10] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum, 1997. [11] D. Van Welden, H. Steendam, and M. Moeneclaey, “Time delay estimation for KSP-OFDM systems in multipath fading channels,” in Proc. 20th Personal, Indoor, Mobile Radio Communications Symp. (PIMRC), Tokyo, Japan, Sep. 2009, pp. 3064–3068. [12] E. de Carvalho and D. T. M. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Apr. 1997, vol. 5, pp. 3593–3596. [13] O. Rousseaux, G. Leus, P. Stoica, and M. Moonen, “Gaussian maximum-likelihood channel estimation with short training sequences,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2945–2955, Nov. 2005. [14] O. Rousseaux, G. Leus, and M. Moonen, “Estimation and equalization of doubly selective channels using known symbol padding,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 979–990, Mar. 2006. [15] R. Cendrillon and M. Moonen, “Efficient equalizers for single and multicarrier environments with known symbol padding,” in Proc. Int. Symp. Signal Processing Its Applications (ISSPA), Aug. 2001, pp. 607–610. [16] H. Steendam, M. Moeneclaey, and H. Bruneel, “An ML-based estimate and the Cramer-Rao bound for data-aided channel estimation in KSP-OFDM,” EURASIP J. Wireless Commun. Netw., 2008, Article DOI 10.1155/2008/186809. [17] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Stat. Soc. Series B, vol. 39, no. 1, pp. 1–38, 1997. [18] X. Ma, H. Kobayashi, and S. C. Schwartz, “EM-based channel estimation algorithms for OFDM,” EURASIP J. Appl. Signal Process., vol. 2004, no. 10, pp. 1460–1477, 2004. [19] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 2001. [20] R. Ravikanth and S. P. Meyn, “Bounds on achievable performance in the identification and adaptive control of time-varying systems,” IEEE Trans. Autom. Control, vol. 44, no. 4, pp. 670–682, Apr. 1999.
Robust Precoding With Bayesian Error Modeling for Limited Feedback MU-MISO Systems Michael Joham, Paula M. Castro, Luis Castedo, and Wolfgang Utschick
Abstract—We consider the robust precoder design for multiuser multiple-input single-output (MU-MISO) systems where the channel state information (CSI) is fed back from the single antenna receivers to the centralized transmitter equipped with multiple antennas. We propose to compress the feedback data by projecting the channel estimates onto a vector basis, known at the receivers and the transmitter, and quantizing the resulting coefficients. The channel estimator and the basis for the rank reduction are jointly optimized by minimizing the mean-square error (MSE) between the true and the rank-reduced CSI. Expressions for the conditional mean and the conditional covariance of the channel are derived which are necessary for the robust precoder design. These expressions take into account the following sources of error: channel estimation, truncation for rank reduction, quantization, and feedback channel delay. As an example for the robust problem formulation, vector precoding (VP) is designed based on the expectation of the MSE conditioned on the fed-back CSI. Our results show that robust precoding based on fed-back CSI clearly outperforms conventional precoding designs which do not take into account the errors in the CSI. Index Terms—Bayesian methods, channel estimation, channel state information, limited rate feedback, vector broadcast channel.
I. INTRODUCTION We consider a MU-MISO system or vector broadcast channel (BC), i.e., a multiple antennas transmitter and several single-antenna receivers. For this setup, dirty paper coding (DPC) schemes designed according to signal-to-interference-plus-noise ratio (SINR) criteria are able to approach the sum capacity [1], [2]. Similar to [3]–[5], these contributions, however, only consider the ideal case where the CSI at the transmitter is perfectly known. In the more practical case, where only an estimate of the CSI is available at the transmitter, the capacity region of the vector BC has not been found yet. First, the application Manuscript received November 17, 2009; accepted May 27, 2010. Date of publication June 07, 2010; date of current version August 11, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Sung Lee. This work was supported by Xunta de Galicia, Ministerio de Educación y Ciencia of Spain, and FEDER funds of the European Union under Grants 09TIC008105PR, TEC2007-68020-C04-01, and CSD2008-00010. Additionally, this work was prepared through an integrated action funded by the Ministerio de Educación y Ciencia of Spain (fund HA2006-0112) and DAAD of Germany (fund D/06/12809). M. Joham and W. Utschick are with the Associate Institute for Signal Processing, Technische Universität München, 80290 Munich, Germany (e-mail: [email protected]; [email protected]). P. M. Castro and L. Castedo are with the Department of Electronics and Systems, University of A Coruña, A Coruña, 15071 Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this correspondence are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2052046
1053-587X/$26.00 © 2010 IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 9, SEPTEMBER 2010
of DPC is questionable, since it is unclear up to now how DPC can be used with erroneous CSI. Second, it is unclear how to systematically include the uncertainties in the SINR criterion (see the discussion in [6] and the attempt in [7] for statistical CSI). As shown in [8], the SINR and MSE achievable regions for MU-MISO systems are tightly related. Additionally, minimum MSE (MMSE) allows for a robust precoder design by considering a conditional expectation of the cost function [9]–[12]. Hence, we focus on the MMSE precoder design. By taking the expectation of the MSE conditioned on the available CSI, robust linear precoding (LP), robust Tomlinson–Harashima precoding (THP), and robust VP evolve from the MMSE designs for error-free CSI of LP in [13], [14], THP in [5], and vector precoding (VP) in [15], respectively. Most of the work on precoding with erroneous CSI was motivated by a time division duplex (TDD) setup, where the transmitter estimates the CSI during the transmission in the opposite direction (e.g., [11]). This approach, however, is difficult due to the need of very good calibration [16]. Contrarily, we focus on the more difficult case where the CSI is obtained by the receivers and fed back to the transmitter. In this case, calibration errors are estimated as being part of the CSI and, therefore, no special problems arise from calibration. Additionally, the feedback of CSI enables precoding in frequency division duplex (FDD) systems, where the transmitter is unable to obtain the CSI during reception, because the channels are not reciprocal. Since the data rate of the feedback channels is limited [17], the CSI must be compressed. Moreover, when the CSI is not perfectly known by the receiver, it is a matter of discussion what kind of information has to be sent from the receiver to the transmitter and the way of recovering it at the transmitter side. In order to properly design robust precoders for the MU-MISO system described in Section II, it is necessary to obtain an adequate statistical characterization of the errors in the fed-back CSI. The following sources of error are considered: channel estimation, truncation (rank reduction), quantization, and feedback channel delay. In the considered system, the observations of pilot symbols sent from all the transmit antennas enable the receivers to estimate their respective vector channels. These estimates are reduced to a low-dimensional representation by projecting them onto a basis depending only on the channel statistics which are assumed to be known also to the transmitter. In Section III-A, the joint design of estimation and rank-reduction based on an MMSE criterion is outlined (see [18]). Since the delayed and truncated estimates are jointly Gaussian distributed with the channels, the analysis of the respective errors follows a conventional MSE approach [19], as shown in Section III-B. In Section III-C, the result obtained for the estimation, rank reduction, and feedback delay errors is extended by the quantization error, where a uniform scalar quantizer is assumed. This suboptimal quantizer has the advantage that it is not necessary to adapt it to changing channel statistics. Moreover, it leads to closed-form expressions for the mean and covariance matrix of the channel conditioned on the delayed, truncated, and quantized channel estimate that are necessary for the robust precoder design presented for the example of robust VP in Section IV. The MMSE receivers and the used training data are discussed in Section V. Simulations are presented in Section VI. 2 identity matrix is denoted by IK and 0K is a -diThe mensional zero vector. We use E[],
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 9, SEPTEMBER 2010
ACKNOWLEDGMENT The authors would like to acknowledge the activity of the Network of Excellence in Wireless COMmunications NEWCOM++ of the European Commission (Contract 216715) that motivated this work.
[21] F. Tufvesson and T. Maseng, “Pilot assisted channel estimation for OFDM in mobile cellular systems,” in Proc. IEEE Veh. Tech. Conf., Phoenix, AZ, May 1997, pp. 1639–1643. [22] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001.
REFERENCES [1] J. A. C. Bingham, “Multicarrier modulation for data transmission, an idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990. [2] Transmission and Multiplexing (TM); Access Transmission Systems on Metallic Access Cables; Very High Speed Digital Subscriber Line (VDSL); Part 2: Transceiver Specification, ETSI TS 101 270-2, Feb. 2001. [3] Digital Audio Broadcasting (DAB); DAB to Mobile Portable and Fixed Receivers, ETS 300 401, Feb. 1995. [4] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb. 1995. [5] R. van Nee, G. Awater, M. Morikura, H. Takanashi, M. Webster, and K. W. Halford, “New high-rate wireless LAN standards,” IEEE Commun. Mag., vol. 37, no. 12, pp. 82–88, Dec. 1999. [6] B. Muquet, Z. Wang, G. B. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixing or zero padding for wireless multicarrier transmissions,” IEEE Trans. Commun., vol. 50, no. 12, pp. 2136–2148, Dec. 2002. [7] H. Steendam and M. Moeneclaey, “Different guard interval techniques for OFDM: Performance comparison,” in Proc. 6th Int. Workshop Multi-Carrier Spread Spectrum (MC-SS), Herrsching, Germany, May 2007, pp. 11–24. [8] T. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [9] L. Deneire, B. Gyselinckx, and M. Engels, “Training sequence versus cyclic prefix—A new look on single carrier communication,” IEEE Commun. Lett., vol. 5, no. 7, pp. 292–294, Jul. 2001. [10] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum, 1997. [11] D. Van Welden, H. Steendam, and M. Moeneclaey, “Time delay estimation for KSP-OFDM systems in multipath fading channels,” in Proc. 20th Personal, Indoor, Mobile Radio Communications Symp. (PIMRC), Tokyo, Japan, Sep. 2009, pp. 3064–3068. [12] E. de Carvalho and D. T. M. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Apr. 1997, vol. 5, pp. 3593–3596. [13] O. Rousseaux, G. Leus, P. Stoica, and M. Moonen, “Gaussian maximum-likelihood channel estimation with short training sequences,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2945–2955, Nov. 2005. [14] O. Rousseaux, G. Leus, and M. Moonen, “Estimation and equalization of doubly selective channels using known symbol padding,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 979–990, Mar. 2006. [15] R. Cendrillon and M. Moonen, “Efficient equalizers for single and multicarrier environments with known symbol padding,” in Proc. Int. Symp. Signal Processing Its Applications (ISSPA), Aug. 2001, pp. 607–610. [16] H. Steendam, M. Moeneclaey, and H. Bruneel, “An ML-based estimate and the Cramer-Rao bound for data-aided channel estimation in KSP-OFDM,” EURASIP J. Wireless Commun. Netw., 2008, Article DOI 10.1155/2008/186809. [17] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Stat. Soc. Series B, vol. 39, no. 1, pp. 1–38, 1997. [18] X. Ma, H. Kobayashi, and S. C. Schwartz, “EM-based channel estimation algorithms for OFDM,” EURASIP J. Appl. Signal Process., vol. 2004, no. 10, pp. 1460–1477, 2004. [19] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 2001. [20] R. Ravikanth and S. P. Meyn, “Bounds on achievable performance in the identification and adaptive control of time-varying systems,” IEEE Trans. Autom. Control, vol. 44, no. 4, pp. 670–682, Apr. 1999.
Robust Precoding With Bayesian Error Modeling for Limited Feedback MU-MISO Systems Michael Joham, Paula M. Castro, Luis Castedo, and Wolfgang Utschick
Abstract—We consider the robust precoder design for multiuser multiple-input single-output (MU-MISO) systems where the channel state information (CSI) is fed back from the single antenna receivers to the centralized transmitter equipped with multiple antennas. We propose to compress the feedback data by projecting the channel estimates onto a vector basis, known at the receivers and the transmitter, and quantizing the resulting coefficients. The channel estimator and the basis for the rank reduction are jointly optimized by minimizing the mean-square error (MSE) between the true and the rank-reduced CSI. Expressions for the conditional mean and the conditional covariance of the channel are derived which are necessary for the robust precoder design. These expressions take into account the following sources of error: channel estimation, truncation for rank reduction, quantization, and feedback channel delay. As an example for the robust problem formulation, vector precoding (VP) is designed based on the expectation of the MSE conditioned on the fed-back CSI. Our results show that robust precoding based on fed-back CSI clearly outperforms conventional precoding designs which do not take into account the errors in the CSI. Index Terms—Bayesian methods, channel estimation, channel state information, limited rate feedback, vector broadcast channel.
I. INTRODUCTION We consider a MU-MISO system or vector broadcast channel (BC), i.e., a multiple antennas transmitter and several single-antenna receivers. For this setup, dirty paper coding (DPC) schemes designed according to signal-to-interference-plus-noise ratio (SINR) criteria are able to approach the sum capacity [1], [2]. Similar to [3]–[5], these contributions, however, only consider the ideal case where the CSI at the transmitter is perfectly known. In the more practical case, where only an estimate of the CSI is available at the transmitter, the capacity region of the vector BC has not been found yet. First, the application Manuscript received November 17, 2009; accepted May 27, 2010. Date of publication June 07, 2010; date of current version August 11, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Sung Lee. This work was supported by Xunta de Galicia, Ministerio de Educación y Ciencia of Spain, and FEDER funds of the European Union under Grants 09TIC008105PR, TEC2007-68020-C04-01, and CSD2008-00010. Additionally, this work was prepared through an integrated action funded by the Ministerio de Educación y Ciencia of Spain (fund HA2006-0112) and DAAD of Germany (fund D/06/12809). M. Joham and W. Utschick are with the Associate Institute for Signal Processing, Technische Universität München, 80290 Munich, Germany (e-mail: [email protected]; [email protected]). P. M. Castro and L. Castedo are with the Department of Electronics and Systems, University of A Coruña, A Coruña, 15071 Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this correspondence are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2052046
1053-587X/$26.00 © 2010 IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 9, SEPTEMBER 2010
of DPC is questionable, since it is unclear up to now how DPC can be used with erroneous CSI. Second, it is unclear how to systematically include the uncertainties in the SINR criterion (see the discussion in [6] and the attempt in [7] for statistical CSI). As shown in [8], the SINR and MSE achievable regions for MU-MISO systems are tightly related. Additionally, minimum MSE (MMSE) allows for a robust precoder design by considering a conditional expectation of the cost function [9]–[12]. Hence, we focus on the MMSE precoder design. By taking the expectation of the MSE conditioned on the available CSI, robust linear precoding (LP), robust Tomlinson–Harashima precoding (THP), and robust VP evolve from the MMSE designs for error-free CSI of LP in [13], [14], THP in [5], and vector precoding (VP) in [15], respectively. Most of the work on precoding with erroneous CSI was motivated by a time division duplex (TDD) setup, where the transmitter estimates the CSI during the transmission in the opposite direction (e.g., [11]). This approach, however, is difficult due to the need of very good calibration [16]. Contrarily, we focus on the more difficult case where the CSI is obtained by the receivers and fed back to the transmitter. In this case, calibration errors are estimated as being part of the CSI and, therefore, no special problems arise from calibration. Additionally, the feedback of CSI enables precoding in frequency division duplex (FDD) systems, where the transmitter is unable to obtain the CSI during reception, because the channels are not reciprocal. Since the data rate of the feedback channels is limited [17], the CSI must be compressed. Moreover, when the CSI is not perfectly known by the receiver, it is a matter of discussion what kind of information has to be sent from the receiver to the transmitter and the way of recovering it at the transmitter side. In order to properly design robust precoders for the MU-MISO system described in Section II, it is necessary to obtain an adequate statistical characterization of the errors in the fed-back CSI. The following sources of error are considered: channel estimation, truncation (rank reduction), quantization, and feedback channel delay. In the considered system, the observations of pilot symbols sent from all the transmit antennas enable the receivers to estimate their respective vector channels. These estimates are reduced to a low-dimensional representation by projecting them onto a basis depending only on the channel statistics which are assumed to be known also to the transmitter. In Section III-A, the joint design of estimation and rank-reduction based on an MMSE criterion is outlined (see [18]). Since the delayed and truncated estimates are jointly Gaussian distributed with the channels, the analysis of the respective errors follows a conventional MSE approach [19], as shown in Section III-B. In Section III-C, the result obtained for the estimation, rank reduction, and feedback delay errors is extended by the quantization error, where a uniform scalar quantizer is assumed. This suboptimal quantizer has the advantage that it is not necessary to adapt it to changing channel statistics. Moreover, it leads to closed-form expressions for the mean and covariance matrix of the channel conditioned on the delayed, truncated, and quantized channel estimate that are necessary for the robust precoder design presented for the example of robust VP in Section IV. The MMSE receivers and the used training data are discussed in Section V. Simulations are presented in Section VI. 2 identity matrix is denoted by IK and 0K is a -diThe mensional zero vector. We use E[],
